a set of equal regular payments with compound interest

Annuities and Loans | HSC Maths Advanced

In short: An annuity is a series of equal regular payments that earn compound interest, like saving into an account every month. A loan works the same way backwards: the amount borrowed is the present value of all its repayments.

Starting point: Years 9 and 10

Building a balance table

An annuity is a series of equal payments made at regular times, with compound interest. Savings plans, superannuation and loans are all annuities.

Suppose you deposit 1000 dollars at the end of each year into an account paying 5% per year.

Year	Start	Interest (5%)	Deposit	End
1	0	0	1000	1000.00
2	1000.00	50.00	1000	2050.00
3	2050.00	102.50	1000	3152.50

After 3 years the account holds 3152.50 dollars. You paid in 3000 dollars, so 152.50 dollars is interest.

A loan table

A loan works the other way. Interest is added each period, then the repayment is taken away.

Borrow 2000 dollars at 1% per month and repay 700 dollars a month:

Month	Start	Interest (1%)	Repayment	End
1	2000.00	20.00	700	1320.00
2	1320.00	13.20	700	633.20
3	633.20	6.33	639.53	0.00

The last payment is smaller: just enough to clear the debt.

Annuities

Regular payments

Deposits: savings
Repayments: loans

Tables

Interest then payment

Future value

FV = M((1+r)ⁿ - 1)/r

Present value

PV = FV / (1+r)ⁿ

Loans

Reducing balance
Repayment formula

Interest factor tables

Value of 1 dollar

Year 12Future and present value formulas

Future value. Deposits of $M$ at the end of each period for $n$ periods. The first deposit earns interest for $n - 1$ periods, the last for none. The total is a geometric series:

$F V = M + M (1 + r) + \dots + M (1 + r)^{n - 1} = M [\frac{( 1 + r ) ^{n} - 1}{r}]$

Check the table: $1000 \times \frac{1.0 5 ^{3} - 1}{0.05} = 3152.50$ .

Present value. Bring the future value back to today:

$P V = \frac{F V}{( 1 + r ) ^{n}} = M [\frac{1 - ( 1 + r ) ^{- n}}{r}]$

Interest factor tables. Exams often give a table of future or present values of an annuity of 1 dollar. Multiply the table value by $M$ . If the future value table shows 6.3081 for $r = 2%$ and $n = 6$ , then six deposits of 500 dollars grow to $500 \times 6.3081 = 3154.05$ dollars.

Sienna's loan. The loan amount is the present value of her repayments, with $r = \frac{0.09}{12} = 0.0075$ and $n = 60$ .

$15000 = M [\frac{1 - 1.007 5 ^{- 60}}{0.0075}] \Rightarrow M \approx 311.38 dollars per month$

She pays $60 \times 311.38 = 18 682.80$ dollars in total, so the interest is about 3683 dollars.

The same result written as one formula is

$M = \frac{P r ( 1 + r ) ^{n}}{( 1 + r ) ^{n} - 1}$

The balance after $n$ repayments. Following the loan rule $A_{n + 1} = A_{n} (1 + r) - M$ from $A_{0} = P$ gives $A_{n} = P (1 + r)^{n} - M [\frac{( 1 + r ) ^{n} - 1}{r}]$ The loan is repaid when $A_{n} = 0$ . To find how long that takes, rearrange to get $(1 + r)^{n}$ alone and use logs. For 20 000 dollars at 1% per month with 400 dollar repayments: $20000 (1.01)^{n} = 40000 (1.0 1^{n} - 1)$ gives $1.0 1^{n} = 2$ , so $n = \frac{ln 2}{ln 1.01} \approx 69.7$ , that is 70 months.

Repayments that never finish. In the first month the interest is $P r$ . If the repayment $M$ is no more than $P r$ , the balance never falls. A 400 000 dollar loan at 0.6% a month has 2400 dollars of interest in month one, so repayments of 2400 dollars or less never repay it.

Total interest = total repaid minus the amount borrowed: $n M - P$ .

Deposits at the start of each period. Each deposit earns one extra period of interest, so multiply the usual future value by $(1 + r)$ .

Withdrawals. Drawing $W$ each month from savings works like a loan: $A_{n} = P (1 + r)^{n} - W [\frac{( 1 + r ) ^{n} - 1}{r}]$ . The money lasts forever only if $W \leq P r$ , the interest earned each month.

Quick summary

An annuity is a set of equal regular payments with compound interest.
Savings tables: add interest, then add the deposit. Loan tables: add interest, then subtract the repayment.
$F V = M [\frac{( 1 + r ) ^{n} - 1}{r}]$ and $P V = \frac{F V}{( 1 + r ) ^{n}}$ .
A loan amount is the present value of its repayments.
With a table of values for 1 dollar, multiply by the payment.

Quiz

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Try 3 questions now

An annuity is:
1. A a single deposit
2. B a set of equal regular payments with compound interest
3. C a type of simple interest
4. D a tax
Show answer
B: a set of equal regular payments with compound interest
Savings plans and loans are annuities.
You deposit 500 dollars at the end of each year at 10%. What is the balance after 2 years?
1. A 1000 dollars
2. B 1050 dollars
3. C 1100 dollars
4. D 1155 dollars
Show answer
B: 1050 dollars
Year 1: 500. Year 2: $500 \times 1.1 + 500 = 1050$ .
A loan of 1000 dollars has 2% interest per month. After one month and a 300 dollar repayment, the balance is:
1. A 700 dollars
2. B 720 dollars
3. C 714 dollars
4. D 680 dollars
Show answer
B: 720 dollars
$1000 \times 1.02 - 300 = 720$ .

Annuities and Loans